# Isometries

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Fix in Consider the real linear map

given by

(a) With respect to the basis B = {1, i, j, k}, find the associated matrix for .

(b) Find the associated matrix A (sub alpha bar) for M (sub alpha bar). Compare with (a).

(c) Compute det(A sub alpha) and det (A sub alpha bar). Interpret.

See attached file for full problem description.

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#### Solution Preview

In order to be able to write them, I will use a and b instead of alpha and beta; I will also use M_a as a notation for M sub a, and conj(a) will mean a bar (the conjugate of a). The problem does not specify, but I assume H means the algebra of quaternions.

(a)our element a in H can be uniquely written as x+y*i+z*j+t*k, where x,y,z,t are ...

#### Solution Summary

This shows how to find associated matrices for a given linear map.

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